General advice, in addition to Bob3160's excellent tutorial:
-Always install or uninstall software as an administrator.
-When installing or uninstalling antiviruses, the replacement install package (or a product CD) should be available for installation prior to uninstalling the other AV.
-Be disconnected from the net while this is going on.
-In the case of Avast, the self-protection module should be disabled (program settings >troubleshooting) before it can be uninstalled from the control panel. The uninstall tool will work in cases where this has not taken place, but will need to be run - as admin - in safe mode.
I would try running the McAfee uninstall tool, then the Avast uninstall utility, rebooting into normal, then reinstalling Avast from scratch. (Then reconnecting to the net for updates.)
This is assuming there is not yet another AV that might have been installed, also. If so, post it. There might be a special procedure or uninstall tool for it. (True of Norton, AVG, Avira....)